Stanford University Department of Mathematics
Math 53, Fall 2007 — Final Exam
Instructor : Samuel Lisi
Date: 10 December 2007
Duration: 3 hours
Family Name :
Given Name(s) :
Student Number :
Your signature:
DO NOT OPEN THIS TEST UNTIL INSTRUCTED TO DO SO.
Instructions:
• Your signature above indicates that you have abided
by the Stanford Honor Code while writing this test.
• There are eight questions. The last part of question 8
is a bonus.
• You may quote theorems from your textbook or from
class if you make an appropriate reference.
Question
1 [10 pts]
2 [10 pts]
3 [15 pts]
4 [15 pts]
• Show all your work.
5 [15 pts]
• No electronic devices of any kind (e.g. calculators,
cell-phones) are allowed.
6 [15 pts]
• There is a table of Laplace transform identities on the
last page of the exam. You may use these identities
without proof, unless the question indicates otherwise.
7 [15 pts]
8 [20 pts]
Total
[100 points]
Marks
Math 53 — Final Exam
1. (a)
Page 1
i. Find a solution to the initial value problem
y 0 = ty 2 ,
y(0) = 1.
ii. What is the interval of existence of this solution?
iii. Is this solution unique? If yes, explain why. If no, provide a second solution.
(b) Find the general solution to the differential equation
2
y 0 + 2ty = e−t sin(t).
[10 pts]
Math 53 — Final Exam
Page 2
[10 pts]
2. Find the integral curve of
α = (sin(x + y) + 2x)dx + (y 2 + sin(x + y))dy
that passes through the point (π, 0). (You may define this curve implicitly.)
Math 53 — Final Exam
Page 3
[15 pts]
3. (a) Solve the initial value problem :
y 00 + 4y = 2 cos(2t),
y(0) = 0, y 0 (0) = 0.
(b) Solve the initial value problem :
y 00 + 4y = 2 cos(2t),
y(0) = 1, y 0 (0) = −1.
Math 53 — Final Exam

2 2 1
Page 4
[15 pts]



.
4. Let A = 
0
2
1


0 0 2
(a) Find the matrix exponential eAt .
(b) Solve the initial value problem :


e2t sin(t)


2t
,
y0 = Ay + 
e


0
 
0
 

y(0) = 
0 .
1
Math 53 — Final Exam
Page 5
0
5. (a) Sketch the phase portrait for y =
[15 pts]
!
−1
2
2
−1
y.
(b) Determine whether 0 is asymptotically stable, stable or unstable.
Math 53 — Final Exam
Page 6
[15 pts]
6. Consider the nonlinear system of differential equations given by :
x0 = 2x(2 − x)
y 0 = y(1 + x2 )
(a) Find the equilibrium solutions.
(b) For each equilibrium solution you found in (a), determine its stability.
Math 53 — Final Exam
Page 7
[15 pts]
7. (a) Find the Laplace transform of the solution to the initial value problem
y 00 + 2y 0 − y = cos(2t),
y 0 (0) = 1, y(0) = −1.
(You do not need to solve for y(t).)
−4
.
(s − 1)2 (s − 3)
The two parts of this question are unrelated.
(b) Find the inverse Laplace transform of
Math 53 — Final Exam
Page 8
[a–c: 10
8. Consider the nonlinear system of differential equations given by :
x0 = 2y
y 0 = −2x − 4x3
(a) Show that (0, 0) is the only equilibrium solution.
(b) What does the linearization at (0, 0) tell us about stability?
(c) Suppose that (x(t), y(t)) is a solution to the nonlinear system. Show that it is an integral
curve of
α = (2x + 4x3 )dx + 2ydy.
(d) [Bonus – 10 pts] Use the fact that α is exact to conclude that (0, 0) is stable, but not
asymptotically stable.
pts]
Math 53 — Final Exam
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Math 53 — Final Exam
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Math 53 — Final Exam
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Page 11
Math 53 — Final Exam
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Page 12
Laplace transform identities
In the following, let f be piecewise continuous of exponential order, and F = L(f ) :
L(ect f (t)) = F (s − c)
L(tf (t)) = −F 0 (s)
1
L(1) =
for s > 0
s
n!
L(tn ) = n+1 for s > 0
s
a
for s > 0
L(sin(at)) = 2
s + a2
s
L(cos(at)) = 2
for s > 0
s + a2
1
L(eat ) =
for s > a
s−a
b
for s > a
L(eat sin(bt)) =
(s − a)2 + b2
s−a
L(eat cos(bt)) =
for s > a
(s − a)2 + b2
n!
L(tn eat ) =
for s > a
(s − a)n+1
If f is piecewise differentiable, and both f and f 0 are of exponential order, then there exists
a > 0 so that :
L(f 0 (t)) = sF (s) − f (0)
for s > a.